In this section, we detail the linearized model and the calculation of second-order photon correlations. First, we begin with a general quartic Hamiltonian for a Kerr resonator with a single pump; we linearize the model to produce a quadratic Hamiltonian, where modes are coupled through the classical amplitudes which can be described by the LLE[1]}. Next, we define the input-output equations for the open quantum system, which we use to calculate the second-order correlation functions in a below-threshold comb. We numerically model the experiment of Fig. 3 from the main text to demonstrate the interface of the LLE with the input-output formalism. We then extend our model to describe photon correlations in the presence of bichromatic coherent light. Finally, we describe how to characterize entanglement (the logarithmic negativity, \(E_N\) ) between different cavity modes starting from the Heisenberg equations.
